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Write the dual of the LPP, given by

Min.: z=2x1 + 3x2 Subject to

2x1 + 3x2 ≤ 30

x1 + 2x2 ≥ 10

x1 - x2 ≥ 0

x1,x2 ≥ 0

Further, without solving primal LPP or dual LPP, check if

(x1*=10/3, x2*=10/3) and

(y1*=0, y2*=5/3, y3*=1/3) are optimal for primal and dual, respectively.

Min.: z=2x1 + 3x2 Subject to

2x1 + 3x2 ≤ 30

x1 + 2x2 ≥ 10

x1 - x2 ≥ 0

x1,x2 ≥ 0

Further, without solving primal LPP or dual LPP, check if

(x1*=10/3, x2*=10/3) and

(y1*=0, y2*=5/3, y3*=1/3) are optimal for primal and dual, respectively.

Solve the following LP by simplex method: Minimise: 2x1- 3x2 Subject to

x1 + x2 ≤ 4

x1 - x2 ≤ 6

x1,x2 ≥ 0

x1 + x2 ≤ 4

x1 - x2 ≤ 6

x1,x2 ≥ 0

A 400-metre medley relay involves four different swimmers, who successively swim 100 metres of backstroke, breakstroke, butterfly and freestyle. A coach has six very fast swimmers whose expected times (in seconds) in the individual events are as follows:

Swimmer Time(in seconds)

Event1 Event2 Event3 Event4

A 65 73 63 57

B 67 70 65 58

C 68 72 69 55

D 67 75 70 59

E 71 69 75 57

F 69 71 66 59

How should the coach assign swimmers to the relay so as to minimise the sum of their times?

Swimmer Time(in seconds)

Event1 Event2 Event3 Event4

A 65 73 63 57

B 67 70 65 58

C 68 72 69 55

D 67 75 70 59

E 71 69 75 57

F 69 71 66 59

How should the coach assign swimmers to the relay so as to minimise the sum of their times?

A project consists of 7 activities with the following relevant information:

Activity Immediate Duration

Predecessor(s) (days)

A - 9

B - 8

C A 5

D C 3

E D 2

F B,E 1

G E,F 14

i) Construct a corresponding network diagram.

ii) Identify a critical path and find the total project duration.

Activity Immediate Duration

Predecessor(s) (days)

A - 9

B - 8

C A 5

D C 3

E D 2

F B,E 1

G E,F 14

i) Construct a corresponding network diagram.

ii) Identify a critical path and find the total project duration.

Let the final table of the LP relaxation of an integer linear programming be as follows:

Cost: 0 2

Basic variables: x4 x1

2 1 1 0 0 solution

x1 x2 x3 x4 x5

0 -1/2 1/2 1 -1/2 3/2

1 3/2 1/2 0 1/2 7/2

0 -2 0 0 -1 7

Use the branch and bound method to find an optimal solution to the original integer linear programming problem.

Cost: 0 2

Basic variables: x4 x1

2 1 1 0 0 solution

x1 x2 x3 x4 x5

0 -1/2 1/2 1 -1/2 3/2

1 3/2 1/2 0 1/2 7/2

0 -2 0 0 -1 7

Use the branch and bound method to find an optimal solution to the original integer linear programming problem.

The following data for the activities of a project are available:

ACTIVITY IMMEDIATE PREDECESSOR DURATION (DAYS) A - 6 B - 4 C - 7 D A,B,C 5 E A,B,C 6 F D 9 G E 4 H F,G 8 I F,G 6 J H 7

Required: i. Draw the network, number the nodes, find the project completion time and calculate total, free and independent floats for the activities ii. When the activity duration of G is doubled, identify the critical path

ACTIVITY IMMEDIATE PREDECESSOR DURATION (DAYS) A - 6 B - 4 C - 7 D A,B,C 5 E A,B,C 6 F D 9 G E 4 H F,G 8 I F,G 6 J H 7

Required: i. Draw the network, number the nodes, find the project completion time and calculate total, free and independent floats for the activities ii. When the activity duration of G is doubled, identify the critical path

A school psychologist believes that a popular new hypnosis technique increases depression. The psychologist collects a sample of 22 students and gives them the hypnosis once a week for two months. Afterwards the students fill out a depression inventory in which their average score was 46.62. Normal individuals in the population have a depression inventory average of 49 with a variance of 169.00. What can the psychologist conclude with an α of 0.01?

a) What is the appropriate test statistic?

b)

Population:

Sample:

c) Compute the appropriate test statistic(s) to make a decision about H0.

(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)

critical value =

; test statistic =

a) What is the appropriate test statistic?

b)

Population:

Sample:

c) Compute the appropriate test statistic(s) to make a decision about H0.

(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)

critical value =

; test statistic =

a sales girl gave a change of 1.15 to a customer instead of 1.25. calculate her percentage error

For the equation y = 5 + 4x^2 + 5x^3

(a) Find the equation for the linear approximation when x = 3.

(b) Find the equation for the quadratic approximation, also when x = 3.

(a) Find the equation for the linear approximation when x = 3.

(b) Find the equation for the quadratic approximation, also when x = 3.

For the equation y = 2 + 3 + 4x^2 + 5x^3

(a) Find the equation for the linear approximation when x = 3.

(b) Find the equation for the quadratic approximation, also when x = 3.

(a) Find the equation for the linear approximation when x = 3.

(b) Find the equation for the quadratic approximation, also when x = 3.